The top two private universities in Japan are Keio and Waseda, and like the famous Yale-Harvard football rivalry, the two schools are always competing against each other fiercely. My son attends a Waseda affiliated high school, and the first thing they taught him was how to sing the school song and cheer loudly for his team against their sworn enemy during the Soukei-sen baseball game. While both schools are very prestigious, I think that Keio has the edge over Waseda when all is said and done. Keio was established by Yukichi Fukuzawa, one of the Founding Fathers of modern Japan, who promoted education as a way for the country to bring the benefits of modernization to every person. (His face adorns the 10,000 yen note today, in a way making him the Ben Franklin of Japan.) Finally, the kanji for Waseda takes away some of its prestige. While the characters for Keio University （慶応大学）are regal, named after the period of imperial rule that came before the Meiji Period, Waseda University (早稲田大学) is less so: the second character in Waseda's name is 稲 ine meaning a rice plant sprouting up, and the third is the kanji for rice paddy 田, which causes Keio students to snicker about how "rural" Waseda is, despite the fact that both schools are in bustling Tokyo.
Now you know more about universities in Japan than you probably needed to know!